Lecture 1
NHMRC Clinical Trials Centre, University of Sydney
gillian.heller@sydney.edu.au
I wish to acknowledge the Ngunnawal people as traditional custodians of the land we are meeting on and recognise any other people or families with connection to the lands of the ACT and region.
I wish to acknowledge and respect their continuing culture and the contribution they make to the life of this city and this region.
I would also like to acknowledge and welcome other Aboriginal and Torres Strait Islander people who may be attending today’s event.
Materials for today’s short course have been inspired by similar short courses given by some of the above colleagues, and some sections have been “borrowed” from their generously shared materials, and from the books that we have written jointly.
| 8:45 - 9am | Tea/coffee on arrival |
| 9 - 10am | Lecture |
| 10 - 10:30am | Morning tea |
| 10:30 - 12pm | Lecture & practical |
| 12 - 1pm | Lunch |
| 1 - 2:30pm | Lecture & practical |
| 2:30 - 3pm | Afternoon tea |
| 3 - 4:30pm | Lecture & practical |
| 4:30pm | Short course end |
| Lecture 1 | Introduction to distributional regression, GAMLSS |
| Lecture 2 | The GAMLSS model, continuous response distributions |
| Lecture 3 | Selecting response distribution, model terms |
| Practical 1 | Plasma data set: select response distribution and covariates |
| Lecture 4 | Model diagnostics |
| Practical 2 | Plasma data set: diagnostics, parameter interpretation |
| Lecture 5 | Discrete response distributions, APTS study |
| Practical 3 | Mixed distributions; speech intelligibility data set |
\begin{align*} y_{i}&\sim\mathcal{N}\left(\mu_i,\sigma^2\right)& \text { independently, for } i=1, \ldots, n \\ \mathbb{E}(y_i)=\mu_i&=\beta_0 + \beta_1 x_{i1}+\ldots+\beta_p x_{ip} \end{align*}
Traditional notation:
\begin{align*} y_i&=\beta_0 + \beta_1 x_{i1}+\ldots+\beta_p x_{ip}+\epsilon_i\\ \epsilon_i&\sim\mathcal{N}\left(0,\sigma^2\right)& \text { independently, for } i=1, \ldots, n \end{align*}
Distribution of y can be non-normal
Relationship between y and x can be nonlinear
Covariate x can affect mean, spread and shape of the distribution
Generalized linear model (GLM)
\begin{align*} y_{i}&\sim \mathcal{E}\left(\mu_i,\sigma\right) \\ g(\mu_i)&=\beta_0 + \beta_1 x_{i1}+\ldots+\beta_p x_{ip} \end{align*}
\mathcal{E} = exponential family (normal, Bernoulli, Poisson, Gamma, inverse Gaussian)
\mu_i=\mathbb{E}(y_i)
\sigma = dispersion parameter
g(\cdot) = link function (monotonic, differentiable)
Generalized additive model (GAM)
\begin{align*} y_{i}&\sim \mathcal{E}\left(\mu_i,\sigma\right) \\ g(\mu_i)&=\beta_0 + s_1(x_{i1})+\ldots+s_p(x_{ip}) \end{align*}
Generalized Additive Models for Location, Scale and Shape (GAMLSS)
\begin{align*} y_{i}&\sim \mathcal{D}\left(\mu_i,\sigma_i,\nu_i,\tau_i\right) \\ g_1(\mu_i)&=\beta_0^{\mu}+s_1^{\mu}(x_{i1})+\ldots+s_p^{\mu}( x_{ip})\\ g_2(\sigma_i)&=\beta_0^{\sigma}+s_1^{\sigma}(x_{i1})+\ldots+s_p^{\sigma}( x_{ip})\\ g_3(\nu_i)&=\beta_0^{\nu}+s_1^{\nu}(x_{i1})+\ldots+s_p^{\nu}( x_{ip})\\ g_4(\tau_i)&=\beta_0^{\tau}+s_1^{\tau}(x_{i1})+\ldots+s_p^{\tau}( x_{ip}) \end{align*}
\mathcal{D} is any distribution with computable log-likelihood and first and second derivatives
\mathcal{D} can have any number of parameters (4 are shown above)
g_k(\cdot) are appropriate link functions
s_j^k(x_{ij}) are smooth functions (e.g. splines) or parametric forms
\begin{align*} y_{i}&\sim\mathcal{N}\left(\mu_i,\sigma^2\right)\\\\ \mathbb{E}(y_i)=\mu_i&=\beta_0 + \beta_1 x_{i} \end{align*}
\begin{align*} y_{i}&\sim \text{GA}\left(\mu_i,\sigma\right) \\\\ \log(\mu_i)&=\beta_0 + \beta_1 x_{i} \end{align*}
\begin{align*} y_{i}&\sim \text{GA}\left(\mu_i,\sigma\right) \\\\ \log(\mu_i)&=\beta_0 + s(x_{i}) \end{align*}
\begin{align*} y_{i}&\sim \text{BCTo}\left(\mu_i,\sigma_i, \nu_i, \tau\right) \\\\ \log(\mu_i)&=\beta_0^\mu + s^\mu(x_{i})\\ \log(\sigma_i)&=\beta_0^\sigma + s^\sigma(x_{i})\\ \log(\nu_i)&=\beta_0^\nu + \beta_1^\nu x_{i}\\ \end{align*}
https://doi.org/10.1186/s12874-020-01021-y
There are different frameworks that enable distributional regression modelling:
generalized additive models for location, scale and shape (GAMLSS),
quantile and expectile regression,
conditional transformation models, and
various other forms.
We will focus on GAMLSS-type models.
2005 seminal paper by Rigby & Stasinopoulos (JRSS)
2005 gamlss package released on CRAN (Stasinopoulos & Rigby)
2023 GAMLSS Working Party formed with the purpose of
gamlss2 version 0.1-0 is currently on GitHub
2025 (we hope) gamlss2 released on CRAN
This document has been written by Nikolaus Umlauf and Thomas Kneib:
Lecture 1